On the Role of Multiply Sectioned Bayesian Networks to Cooperative Multiagent Systems

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1 On the Role of Multiply Sectione Bayesian Networks to Cooperative Multiagent Systems Y. Xiang University of Guelph, Canaa, V. Lesser University of Massachusetts at Amherst, USA, Abstract We consier a common task in multiagent systems where agents nee to estimate the state of an uncertain omain so that they can act accoringly. If each agent only has partial knowlege about the omain an local observations, how can the agents accomplish the task with a limite amount of communication? Multiply sectione Bayesian networks (MSBNs) provie an effective an exact framework for such a task but also impose a set of constraints. Are there simpler frameworks with the same performance but with less constraints? We ientify a small set of high level choices which logically imply the key representational choices leaing to MSBNs. The result aresses the necessity of constraints of the framework. It facilitates comparisons with relate frameworks an provies guiance to potential extensions of the framework. (Keywors: multiagent system, ecentralize interpretation, communication, organization structure, uncertain reasoning, probabilistic reasoning, belief network, Bayesian network) I. INTRODUCTION As intelligent systems are being applie to larger, open an more complex problem omains, many applications are foun to be more suitably aresse by multiagent systems [25], [27]. Consier a large uncertain problem omain populate by a set of agents. The agents are often charge with many tasks etermine by the nature of the application. One common task is to estimate what is the true state of the omain so that they can act accoringly. Such a task, often referre to as istribute interpretation [5], arises in many applications of multiagent systems incluing equipment/process troubleshooting, builing/area surveillance, battle fiel/isaster situation assessment, an istribute esign. We can escribe the omain with a set of variables. Some variables are not irectly observable hence their values can only be inferre base on observations of other variables an backgroun knowlege about their epenence relations. Furthermore, each agent has only a partial perspective of the problem omain. That is, each agent only has knowlege about a subomain, i.e., about the epenence among a subset of omain variables, an can only observe an reason within the subomain. The agents may be evelope by ifferent esigners an the subomain knowlege may be private to their esigners. Hence, maintaining the privacy of the agents while they are cooperating may be esirable. In the case of a single agent, the task of estimating the state of the omain can be achieve by representing the omain knowlege in a Bayesian network (BN) [2] an by performing probabilistic inference using the BN given the agent s observations. However, as multiple agents are cooperating on the task, a set of new issues arise: How shoul the omain be partitione into subomains? How shoul each agent represent its knowlege about a subomain? How shoul the knowlege of each agent relate to that of others? How shoul the agents be organize in their activities? What information shoul they exchange an how, in orer to accomplish their task with a limite amount of communication? Can they achieve the same level of accuracy in estimating the state of the omain as that of a single centralize agent? Multiply sectione Bayesian networks (MSBNs) [28] provie one solution to these issues. An MSBN consists of a set of interrelate Bayesian subnets each of which encoes an agent s knowlege concerning a subomain. Agents are organize into a hypertree structure such that inference can be performe in a istribute fashion while answers to queries are exact with respect to probability theory. Each agent only exchanges information with ajacent agents on the hypertree, an each pair of ajacent agents only exchange their beliefs on a set of share variables. Both local inference within an agent an communication among all agents are efficient when the agent subnets are sparse. Therefore, MSBNs provie a framework in which multiple agents can estimate the state of a omain effectively with exact an istribute probabilistic We shall use the term effective to mean efficient computation when agent subnets are sparse.

2 2 y z g5 e z2 _ w9 i 4 x v7 y v4 g g7 n s x5 t9 z4 _ t8 l 8 v g3 x3 w6 2 w _ u y2 t2 r o 3 a2 6 n o 7 p q b2 g4 U c U g2 b a f g _ v5 z3 t i w7 k _ s g8 g9 q s2 5 p t6 U 3 w8 v6 z t t3 k h2 i2 f w5 w2 _ t4 x4 e2 U 2 z5 t5 t7 j2 _ y4 q2 9 _ l2 _ m2 g6 o2 U 4 n2 _ Fig.. A igital system. inference. In principle, the framework allows unboune number of agents as well as allows agents to join an leave ynamically. Are there simpler alternatives that can achieve the same performance? In other wors, are the technical constraints of MSBN necessary? For example, the hypertree organization of agents prevents an agent from communicating irectly an arbitrarily with another agent. Is this necessary? The agent interface is require to satisfy a conition calle -sepset (etaile in the paper). Is it necessary? In this work, we aress these issues. We show that given some reasonable funamental choice/assumptions, the key constraints of an MSBN, such as a hypertree structure an a -sepset agent interface, follow logically. In particular, we ientify the choice points in the formation of the MSBN framework. We term the funamental choices as basic commitments (BCs). Given the BCs, other technical choices are entaile. Hence, an MSBN or some equivalent follows once we amit the BCs. The contributions of this work are the following: First, the analysis provies a high-level (vs. technical level) escription of the applicability of MSBN an aresses issues regaring the necessity of major MSBN representational constraints. Secon, the results facilitate comparison with alternative frameworks. Thir, the analysis provies a guieline for extensions or relaxations of the MSBN framework as to what can or cannot be trae off. In Section II, we briefly overview the MSBN framework with representational choices summarize. Each remaining section ientifies some BCs an erives implie choices. II. OVERVIEW OF MSBNS A BN [2] is a triplet where is a set of omain variables, is a DAG whose noes are labele by elements of, an is a joint probability istribution (jp) over, specifie in terms of a istribution for each variable conitione on the parents of in. An MSBN [33], [28] is a collection of Bayesian subnets that together efine a BN. For instance, suppose that a piece of equipment consists of multiple components built by ifferent esigners. As a small example, Figure shows a piece of igital equipment mae out of five components! ". Each box in the figure correspons to a component an contains the logical gates an their connections with the input/output signals of each gate labele. A set of five agents, #$ %&! " respectively associate with a component ', cooperate to monitor the system an trouble-shoot it when necessary. Each agent #( is responsible for a particular component ), which is likely being evelope by the esigner of the component. When a gate is enclose in exactly one box, the gate is physically locate in the corresponing component an is logically known only to the agent responsible for the component. On the other han, when a gate is enclose in more than one box, the gate is physically locate in only one of the components but is logically known to all the corresponing agents. For

3 3 Fig. 2. The subnet for. Fig. 3. The subnet for. example, the AND gate is known only to #, the OR gate is known to both # an #, an the signal is known to #, # an #. The knowlege of an agent about its assigne component can be represente as a BN, calle a subnet. The subnet for agent # (responsible for component ) is shown in Figure 2 an that for # is shown in Figure 3. Each noe is labele with a variable name. Only the DAGs of the subnets are shown in the figures with the conitional probability istribution for each variable omitte. As mentione above, an agent not only knows all evices locate in its assigne component, but also knows some evices that are physically locate in other interfacing components. Hence, each subnet encoes the agent s knowlege on both types of evices. The five subnets (one for each component) collectively efine an MSBN, which form the core knowlege of the multiagent system. Base on this knowlege an limite observations, agents can cooperate to estimate whether the system is functioning normally, an if not, which evices are likely to be responsible. For instance, suppose that the gates (in ) an (in ) in Figure break own an prouce incorrect output. Some outputs ownstream are also affecte. Equipment inputs an correct evice outputs are shown in Figure by an. Incor-

4 4 rect outputs are shown unerline. Through limite local observation (each agent observes the values of 3 to 4 signals incrementally) an communication (two rouns), agents can ientify the two faulty gates [3], [28] cor- rectly (with " # an! ). Subnets in an MSBN are require to satisfy certain conitions. To escribe these conitions, we introuce the terminologies first. Let $ %$ '& be two graphs (irecte or unirecte). an are sai to be graph-consistent if the subgraphs of an spanne by )( are ientical. Given two graphconsistent graphs ( %$ '&, the graph * %$ *+$ is calle the union of an, enote by )*. Given a graph %$, a partition of into an such that )* % an ( -, /., an subgraphs of spanne by $ '&, is sai to be sectione into an. See Figure 4 for an example. Note that if an are f e f e G c c a b a b a g h g G h G i b Fig. 4. The graph in (a) is sectione into an in (b). is the union of2 an. sectione from a thir graph, then an are graphconsistent. The union of multiple graphs an the sectioning of a graph into multiple graphs can be similarly efine. Graph sectioning is useful in efining the epenence relation between variables share by agents. It is use to specify the following hypertree conition which must be satisfie by subnets in an MSBN: Definition : Let %$ be a connecte graph sectione into subgraphs 3 ($ %$ 54. Let the subgraphs be organize into an unirecte tree 6 where each noe is uniquely labele by a ( an each link between 87 an :9 is labele by the non-empty interface #72( 9 such that for each an ;, <( = is containe in each subgraph on the path between $ an >= in 6. Then 6 is a hypertree over. Each ( is a hypernoe i (a) (b) an each interface is a hyperlink. Figure 5 illustrates a hypertree for the igital system, where an are shown in Figures 2 an 3. The Fig. 5. G G 4... G g, g 7, g 8, g 9, i, k, n, o, p, q, r, t 2, y 2, z 2, z4 G G 3 The hypertree for the igital equipment monitoring system. hypertree represents an organization of agent communication, where variables in each hypernoe are local to an agent an variables in each hyperlink are share by agents. Agents communicate in an MSBN by exchanging their beliefs over share variables. We use noes an variables interchangeably when there is no confusion. Noes share by subnets in an MSBN must form a -sepset, as efine below: Definition 2: 2 Let be a irecte graph such that a hypertree over exists. A noe containe in more than one subgraph with its parents in is a -sepnoe if there exists at least one subgraph that contains. An interface? is a -sepset if is a -sepnoe. The interface between an contains 3 variables inicate in Figure 5. The corresponing noes in Figures 2 an 3 are unerline. It is a -sepset because these variables are only share by an, an each variable has all its parents containe in one of them. For instance, the parents of ( an A ) are all containe in, while those of (, CB an ) are containe in both an (see Figures 2 an 3). The structure of an MSBN is a multiply sectione DAG (MSDAG) with a hypertree organization: Definition 3: A hypertree MSDAG D, where each is a DAG, is a connecte DAG such that () there exists a hypertree 6 over, an (2) each hyperlink in 6 is a -sepset. Note that although DAGs in a hypertree MSDAG form a tree, each DAG may be multiply connecte. A loop in a graph is a sequence of noes %E such that the first noe is ientical to the last noe an there is a link Note that this efinition is an extension of earlier efinitions for -sepset, such as that in [28], to the most general case.

5 5 Fig. 6. f e c G a b a b l n m A MSDAG with multiple paths across local DAGs. (not necessarily in the same irection) between each pair of noes ajacent in the sequence. Such a loop is also referre to as an unirecte loop. A DAG is multiply connecte if it contains at least one (unirecte) loop. Otherwise, it is singly connecte. For example, in Figure 2 has two loops. One of them is!. Hence, is multiply connecte. in Figure 3 has several loops an is also multiply connecte. Moreover, multiple paths may exist from a noe in one DAG to another noe in a ifferent DAG after the DAGs are unione. For instance, in Figure 6, there are several (unirecte) paths from noe E in to noe in. There is one path going through noes, an ; an another path goes through,, an. Each path goes across all three DAGs. An agent s quantitative knowlege about the strength of epenence of a variable on its parent variables can be encoe as a conitional probability istribution!. 2! is a special case of a potential over 3 4 *. A potential over a set of variables is an non-negative istribution of at least one positive parameter. For instance, the following table illustrates a potential over two variables an its parent. It G TABLE I A POTENTIAL OVER A SET OF VARIABLES THAT REPRESENTS A PROBABILITY DISTRIBUTION o j k j k ' & & & & & may represent the probability istribution 2 in the last column. One can always convert into 2 by iviing each potential value with a proper sum: an operation terme normalization. For instance, after iviing the first two values in by their sum 4, we obtain the probability values # an &<. After iviing the last two g h G 2 i values in by their sum, we obtain the probability values # & an &< &. Hence, ' contains the same information as 2 with the flexibility of not having to perform the normalization until it is neee. A uniform potential is one with all its potential values being. An MSBN is then efine as follows. Uniform potentials are use to ensure that quantitative knowlege about the strength of epenence of a variable on its parent variables will not be oubly specifie for the same variable. Definition 4: An MSBN is a triplet!. D is the omain where each is a set of variables. D (a hypertree MSDAG) is the structure where noes of each DAG ( are labele by elements of. Let be a variable an be all the parents of in. For each, exactly one of its occurrences (in a containing 3!4 * ) is assigne 2!, an each occurrence in other DAGs is assigne a uniform potential. #" ' is the jp, where each is the prouct of the potentials associate with noes in. A triplet ' is calle a subnet of. Two subnets an = are sai to be ajacent if an = are ajacent on the hypertree MSDAG. MSBNs provie a framework for the task of estimating the state of an uncertain omain in cooperative multiagent systems. Each agent hols its partial perspective (a subnet) of a omain, reasons about the state of its subomain with local observations an through limite communication with other agents. Each agent may be evelope by an inepenent esigner an the internals of an agent (agent privacy) are protecte. Agents can acquire observations in parallel while their beliefs about the states of iniviual subomains are consistent with observations acquire by all agents. For the igital system example, each component is assigne an agent # in charge of the subnet an its local computation. The representational choices of MSBNs are summarize below, where the most important ones are 3 an 6. ) Each agent s belief is represente by Bayesian probability. 2) The omain is ecompose into subomains. For each pair, there exists a sequence of subomains such that every pair of subomains ajacent in the sequence shares some variables. 3) Subomains are organize into a (hyper)tree structure where each hypernoe is a subomain, an each hyperlink represents an non-empty set of share variables between the two hypernoes such that variables share by any two hypernoes are also share by each hypernoe on the path between

6 6 them. 4) The epenency structure of each subomain is represente by a DAG. 5) The union of DAGs for all subomains is a connecte DAG. 6) Each hyperlink is a -sepset. 7) The joint probability istribution can be expresse as in Definition 4. Below we ientify a set of BCs leaing to these choices. III. ON COMMUNICATION GRAPHS We use uncertain knowlege, belief an uncertainty interchangeably, an make the following basic commitment: BC : Each agent s belief is represente by Bayesian probability. It irectly correspons to the choice of Section II. We shall use coherence to escribe any assignment of belief consistent with the probability theory. We consier a omain of variables populate by cooperative agents # #. Each # has knowlege over, calle the subomain of #. For example, in equipment monitoring, each correspons to a component incluing all its evices an their input/output signals. Although not require in theory, practically it is assume whenever ( =,., the intersection is small relative to an =. From BC, the knowlege of # is a probability istribution over, enote by '. To minimize communication, we allow agents to exchange only their beliefs on share variables (BC 2 below). We take it for grante that for agents to communicate irectly, ( <= must be nonempty. Note that BC 2 oes not restrict the orer nor the number of communications. BC 2: # an # = can communicate irectly only with #( <=. We refer to ( <= as a message an refer to irect communication as message passing. We emphasize that the funamental property of message passing (as use in this paper) is that the messages normally reveal only partial information known to the sener. In other wors, neither a single message nor all the messages from a sener collectively isclose all the information that the sener has. Paths for message passing can be represente by a communication graph (CG): In a graph with noes, associate each noe with an agent #( an label it by. Connect each pair of noes an = by a link labele.. Figure 7 shows the communication graph of the multiagent system for monitoring the igital system from Figure. The subomains an their intersections are shown below: by? ( = if?, V I, V 4 I I,2,2 V 3 2 I Fig. 7. The communication graph of the multiagent system for monitoring the igital system. I 2, ; B B A A A A 4 3 A 4 3 ; B A %E A! 4 %E CB! A)B! 4 CB!! A!! B!!! 4? 3 %E! 4? 3 4? 3 B! 4? 3 A! 4? 3! ; B A! 4 In fact, a communication graph is an application of a general class of graphs calle junction graphs [9]. Although our focus is on communication graphs, many of their relevant properties are intrinsic to all junction graphs. Therefore, we shall escribe these properties in terms of junction graphs whenever it is appropriate. Definition 5 efines junction graphs formally. We use to enote the power set of a set. Definition 5: A junction graph is a triplet %$. is an non-empty set calle the generating set. is a subset of such that *. Each element of is calle a cluster. $ is efine as $ 3! (" #, ($ -. 4, 2,4 V V

7 7 where each unorere pair is calle a separator between the two clusters an, an is labele by the intersection (". In Figure 7, the generating set is the set of all variables in the igital system omain. Each cluster correspons to one component subomain. The separators in a CG represent all potential paths for message passing among agents. As the belief of one agent can influence the belief of another agent through a thir agent, CG also represents all potential paths for inirect communications. Each agent s belief shoul potentially be influential in any other, irectly or inirectly. Otherwise the system can be split into two. Hence, CG is connecte. We summarize this in Proposition 6. It is equivalent to Choice 2 in Section II. Recall that BC stans for basic commitment. Proposition 6: Let be the communication graph over that observes BC an BC 2. If each agent s belief can in general influence that of each other agent through communication, then is connecte. A CG contains all possible paths for agent communication. If we remove some separators from a CG, the agent communication is effectively restricte to a proper subset of potential paths. The resultant graph is a cluster graph as efine below. Definition 7: Let %$ be a junction graph an $ $. Then %$ is a cluster graph over. Note that a junction graph is also a cluster graph, but a cluster graph may not be a junction graph. Cluster graphs of a CG represents alternative organizations for agent communication, which will be stuie in the next section. IV. ON HYPERTREE ORGANIZATION A. Classification of loops The ifficulty of coherent inference in multiply connecte graphical moels (those with loops) of probabilistic knowlege is well known an many inference algorithms have been propose. Those base on message passing, e.g., [2], [4], [], [23], [4], all convert a multiply connecte network into a tree. However, no formal arguments can be foun, e.g., in [2], [9], [8], [3], which emonstrate convincingly that message passing cannot be mae coherent in multiply connecte networks. 3 This leaves the question whether it is impossible to construct In fact, this issue has never been raise openly to the authors knowlege. Pearl [2] explaine that his algorithm for message passing in tree-structure BNs woul not work correctly in multiply connecte BNs because the assumptions that lea to the algorithm woul not hol. He i not, however, treat the issue in general. In fact, an empirical stuy [7] has been performe recently to apply to multiply connecte BNs for approximate inference. such a metho or the metho remains to be iscovere, uner the constraints that each noe in the network is associate with only a local istribution an it is never passe to a central location for manipulation. The answer to this question ties closely to the necessity of the hypertree organization of agents as specifie in Definition 3 an restate as the choice 3 in Section II. This tie can be seen by noting that the hypertree in Definition 3 is isomorphic to a subgraph of the communication graph of the same multiagent system: A one-to-one mapping exists between hypernoes in Definition 3 an noes in. Each hyperlink in Definition 3 is a link in but the converse is not true. Compare Figures 5 an 7 as an example. In what follows, we show that in general, coherent message passing is impossible in general, multiply connecte CGs. The result formally establishes the necessity of hypertree structure for uncertain omain state estimation by multiagent message passing. We first classify loops on a cluster graph as follows: Definition 8: Let be a cluster graph over an be a loop in. If there exists a separator on that is containe in every other separator on, then is a egenerate loop. Otherwise, is a nonegenerate loop. A egenerate loop is a strong egenerate loop if all separators on are ientical. Otherwise, is a weak egenerate loop. An nonegenerate loop is a strong nonegenerate loop if ( -., where is over every separator on. Otherwise, is a weak nonegenerate loop.,e b,c, a,b b b,c, a c,f,g (a) a,e e c,e (c),e,i a,b,f b,f,i b,c,,i b,c,,f a,f c,f,f,h,h,g,h a,e,f e,f c,e,f (b) () Fig. 8. Cluster graphs where each cluster is shown as an oval an each separator is shown in a box. In Figure 8, all loops in (a) an (b) are egenerate. The loops in (a) are strong egenerate loops, an the loop in (b) is a weak egenerate loop. The loops in (c) an () are nonegenerate. The loop in (c) is a strong nonegenerate loop, an that in () is a weak nonegenerate loop. In general, a cluster graph can contain both types of loops an can contain strong an weak loops for each type.

8 8 B. Nonegenerate loops We show that when nonegenerate loops exist, messages are uninformative. No matter how messages are manipulate or route, they cannot become informative an it becomes impossible to make message passing coherent. b a c (a) a,b b a b,c, c a,c (b) Fig. 9. (a) A epenence structure over four variables. (b) The junction graph for the variables in (a). Consier a omain with the epenence structure in Figure 9 (a) where %E are binary (i.e., 3 4, 3 4, an so on). It is populate by three agents # ( '& ) with 3 4, 3 %E4 an 3 %E 4. Although the system appears trivial, it will be expane to arbitrary complexity below. Figure 9 (b) is the communication graph. The local knowlege of agents are, %E an %E, respectively. We assume that their beliefs are initially consistent, namely, the marginal istributions satisfy, (, an E $ E. Due to BC 2, message passing cannot change any agent s belief. We refer to the system as Mas3 (meaning a multiagent system of 3 agents). Any given, %E an %E subject to the above consistency is calle an initial (belief) state of Mas3. Suppose that # observes. If the agents can upate their beliefs coherently, their new beliefs shoul be, %EC an %E. For #, %E can be obtaine locally. However, for # an # to upate their beliefs, they must rely on the message sent by # to # an the message EC sent by # to #. In the following, we show that # an # cannot upate their beliefs coherently base on these messages. Before the general result, we illustrate with a particular initial state. From Figure 9, we can inepenently specify,, EC, an %E as follows: < E E < %E < %E < %E %E From these, we efine an initial state which is consistent: < %E EC %E %E %E where %E ' EC. After is observe by #, its messages are < an EC ". Consier next a ifferent initial state that iffers from %E with %E as follows: < %E < %E && %E & by replacing Note that < < %E %E, %E, but an %E %E. After is ob- an EC are com- serve, if the messages pute, they are foun to be ientical to those obtaine from the state. That is, the messages are insensitive to the ifference between the two initial states. As the consequence, the new beliefs in # an # will be ientical in both cases. Shoul the new beliefs in both cases be ifferent? Using the coherent probabilistic inference, the new belief is obtaine from, an is obtaine from. The ifference is significant. We now show that the above phenomenon is not acciental. Without losing generality, we assume that all istributions are strictly positive. Lemma 9 says that for infinitely many ifferent initial states of agent #, its messages to # an #, however, are ientical. Lemma 9: Let be a strictly positive initial state of Mas3. There exists an infinite set. Each element is an initial state of Mas3 ientical to in,, %E such that the message an EC but istinct in prouce from is ientical to that prouce from, an so is the message E. Proof: We enote the message component C from state by. We enote the message component from by <. < can be expane as & & < &!! Similarly, the message component E < can be expane as E < &" & # #

9 ( By assumption,, %E %E an %E %E but %E, %E. If agent # at generates the ientical messages an E EC (conclusion of the lemma), then %E must be the solution of the following equations:!!!!! %E has four inepenent parameters but! Because is constraine by only two equations, it has infinitely many solutions. Each solution efines an initial state of Mas3 that satisfies all conitions in the lemma. Lemma says that with the same ifference in initial states, a coherent inference will prouce istinct results from Mas3. Lemma : Let an be strictly positive probability istributions over the DAG of Figure 9 such that they are ientical in, an EC but istinct in %E. Then is istinct from in general. Proof: The following can be obtaine from an : %E %E %E () %E (2) where %E is use because is ientical with in, an E. If %EC, %EC (which we shall show below), then in general,. We have!" # $ &%$ % $ % ' $ % (" # % ) Because %E, that %E, %E, in general, it is the case %EC. We conclue with the following theorem: Theorem : Message passing in Mas3 cannot be coherent in general, no matter how it is performe. Proof: By Lemma 9, an EC are insensitive to the initial states an hence the posteriors (e.g., ) compute from the messages cannot be sensitive to the initial states either. However, by Lemma, the posteriors shoul be ifferent in general given ifferent initial states. Hence, correct belief upating cannot be achieve in Mas3. Note that the non-coherence of Mas3 is ue to its nonegenerate loop. From Eqs.() an (2), correct inference requires %EC. To pass such a message, a separator must contain 3 %E4, the intersection between 9 an *. The nonegenerate loop signifies the splitting of such a separator (into separators 3 4 an 3 E4 ). The result is the passing of marginals of %EC (the insensitive messages) an ultimately the incorrect inference. We can generalize this analysis to an arbitrary, strong nonegenerate loop of length 3 (the loop length of Mas3), where each of E is a set of variables. The result in Lemmas 9, an Theorem can be similarly erive. We can further generalize this analysis to an arbitrary, strong nonegenerate loop of length *,+. By clumping *.- ajacent subomains into one big subomain, the loop is reuce to length 3. Any message passing among the /- subomains can be consiere as occurring in the same way as before the clumping but insie. Now the above analysis for an arbitrary strong nonegenerate loop of length 3 applies. Furthermore, the result can be generalize to an arbitrary, weak nonegenerate loop of length *,. Let 2 the separator between an ( be *8- &, the separator between an be, an 9 ;:. Let the potential of each cluster be =< ( 9 =< B9 =< B9 B9 may be ifferent ue to an observation on D9 available to $ but not to )=. This is possible when the object associate with is physically locate with the agent associate with an is only logically known to the agent associate with 8=. Message passing in can be consiere as inepenently passing of two message streams, one etermine by ><?E9 9 an one etermine by =< B9. For example, the message from to is F?G9 9 B9 where >?H9 9 can be obtaine by marginalization of F?G9 9. The first message stream accoring to =< (?I9 9 is equivalent to the message passing in a strong nonegenerate loop where each cluster (@?E9. Accoring to the above analysis, belief upating cannot be achieve by message passing in. The secon message stream relative to

10 4 =< B9 is straightforwar. It in general has no impact on the first message stream. Therefore, belief upating cannot be achieve by message passing in in general. To summarize, the ifficulty will arise whenever a cluster graph contains nonegenerate loops, whether they are strong or weak. This is state in the following Corollary. Corollary 2: Message passing in a cluster graph with nonegenerate loops cannot be coherent in general, no matter how it is performe. C. Degenerate loops In a strong egenerate loop, all subomains share the same separator an it is straightforwar to pass the message coherently. Furthermore, the coherent message passing can be performe with any one separator omitte. That is, it can be performe with the loop cut open into a chain. Next, consier a weak egenerate loop where separators are not all ientical, but there exists a separator that is containe in each other separator. Let the loop be where *. Let the clusters connecte by be an. There are two paths between an, 2 The message that can be passe from to along is a potential % Because the message passe along any separator, potential? can be expresse as a which contains, the path is reunant: the same information can be propagate through the other path. Therefore, whether or not coherent message passing is achievable in a weak egenerate loop can be etermine using the cluster chain obtaine by breaking the loop at. For example, whether coherent message passing is achievable in the cluster graph in Figure (a) can be etermine by eleting the separator 3 4 to obtain (b). Similarly, whether coherent message passing is achievable in the cluster graph in Figure (a) can be etermine by eleting a separator 3 4 to obtain (b). Coherent message passing is achievable in Figure (b) using any well-known methos [4], [9], [24]. Hence, it is also achievable in (a) because one can always ignore the existence of the separator that is omitte in (b). On the other han, coherent message passing is not achievable in general in Figure (b) as exemplifie in Figure 2. The potential of each cluster is shown in terms Fig.. Fig.. a,b,v v,e,v a,b,v,e,v b,v,v b,v,v b,c,v c,v c,,v b,c,v c,v c,,v (a) (b) The weak egenerate loop in (a) is broken into a chain in (b). a,b,u u a,b,u u,e,u u,e,u u u c,,u c,u b,c,u c,,u c,u b,c,u (a) (b) The weak egenerate loop in (a) is broken into a chain in (b). of a single non-zero probability value (with zero being the value of other probabilities). Clearly, the message on the separator 3 4 is /& /& for each corresponing cluster. The message on the separator 3 E is E & & &. Therefore, no matter how message passing is performe, none of the cluster potentials will change. However, accoring to the cluster 3 E 4, we have & &, but accoring to the cluster 3 4, we have &. Because the separator between the clusters 3 4 an 3 %E 4 in (a) cannot help solve the problem, coherent message passing is also not achievable in Figure (a).

11 P(a=,b=,u=)=. a,b,u u,e,u P(=,e=,u=)=. Fig. 2. u P(c=,=,u=)=. c,,u c,u b,c,u P(b=,c=,u=)=. Each pair of ajacent clusters are consistent. The key conclusion here is that the loopy structure of a weak egenerate loop is insignificant just as that of a strong egenerate loop, in the sense that whether it provies support to coherent message passing can be stuie reliably from a erive chain structure. Hence, a cluster graph with only egenerate loops can always be treate by first breaking the loops at appropriate separators. The resultant is a cluster tree. With the unerstaning of the properties of ifferent types of loops, we now make a choice on the organizational structure for agent communication. Given any connecte graph, its connecte spanning subgraphs (containing the same set of noes as ) with the minimum number of links are trees. That is, trees are the simplest (with the minimum number of links) subgraphs that retain connecteness. Simplicity is conuctive to efficiency. Consier a weak egenerate loop in a CG, where a separator is containe in every other separator. If we use the loopy communication organization, there are two information channels between any two clusters in the loop: one through an one through the other path in the loop. Because each separator in the other path is a superset of (by efinition of weak egenerate loop), from BC 2, the information capacity of the path through is inferior to the other path. This implies that not all messages can be passe equivalently from both paths. Hence, agents must select the path carefully epening on the content of their messages. Clearly, this requires more sophisticate computation an coorination than what woul be require in a tree organization. We therefore prefer a simpler organization of agents when egenerate loops exist in the CG: BC 3: A simpler agent organization (as a subgraph of the communications graph) is preferre. From BC 3, a tree organization follows. This is summarize in the following proposition, which implies the choice 3 in Section II. Proposition 3: Let a multiagent system be one that observes BC through BC 3. Then a tree organization of agents shoul be use. Proposition 3 amits many tree organizations. Jensen [9] showe that coherent message passing may not be achieve with just any tree. In particular, if two subomains an = share a subset? of variables but? is not containe in every subomain on the path between them in the tree, then coherent message passing is not achievable. In fact, the cluster tree in Figure (b) suffers precisely this problem. To ensure coherent message passing, the tree must be a junction tree, where for each pair of an =, ( = is containe in every subomain on the path between an =. Note the similarity between a junction tree an a hypertree in Definition. Hence, we have the following proposition: Proposition 4: Let a multiagent system be one that observes BC through BC 3. Then a junction tree organization of agents must be use. V. ON SUBDOMAIN SEPARATORS Given the commitment to a (hyper) junction tree organization, it follows that each separator must be chosen such that the message over it is sufficient to convey all the relevant information from one subtree to the other. Let enote the set of variables in the separator, enote the union of all subomains of one subtree inuce by the separator excluing, an enote the union of all subomains of the other subtree excluing. By BC 2, is the only information that can be irectly communicate between an. Note that because we are concerne with as the separator between an, we can safely ignore the fact that (or ) is istribute among multiple agents. We consier the conition uner which the messages between * an * through are sufficiently informative to ensure coherent message passing. Suppose is associate with a potential an with %. For the isjoint sets,, an of variables, enote? [2] if an are conitionally inepenent given. Using the notation, if? hols, then the joint istribution an local istributions, satisfy where. Now suppose some variables in are observe an is upate into. To upate in, pass the message from to, an replace in by

12 2 The message passing is coherent because What if? oes not hol? Consier the epenence structure in Figure 9 (a). If a Bayesian network is efine with the epenence structure, an constructe by the chain rule as then in general,? %E EC %E %E is 3 E 4 oes not hol. It has been shown in Lemmas 9, an Theorem that passing a message over from cluster 3 %E 4 to 3 4 cannot prouce correct posterior in general. The following proposition summerizes the above analysis. Proposition 5: Let, an be isjoint sets of variables with efine. Let * be associate with, an * be associate with. ) If? hols, then message passing can be performe coherently by passing a potential over between an. 2) If? oes not hol, then message passing cannot be performe coherently in general by passing a potential over between an. To conclue, when the separator reners the two subtrees conitionally inepenent, if new observations are obtaine in one subtree by the corresponing agents, coherent belief upate of agents in the other subtree can be achieve by simply passing the upate istribution on the separator. On the other han, if the separator oes not rener the two subtrees conitionally inepenent, passing only the separator istribution will not be coherent in general. Hence, we have the following proposition: Proposition 6: Let a multiagent system be one that observes BC through BC 3. Then each separator in a tree organization must rener subomains in the two inuce subtrees conitionally inepenent. This commitment requires the problem omain to be partitione among agents such that intersections of subomains form conitional inepenent separators in a hypertree organization. VI. CHOICE ON SUBDOMAIN REPRESENTATION Given a subomain, the number of parameters to represent the belief of # through a potential over is exponential on the carinality 5. Graphical moels allow such belief to be compactly represente. We focus on DAG moels as they are the most concise, with the unerstaning that other moels such as ecomposable Markov networks [2], [3], [34] or chain graphs [3] may also be use. This correspons to the choice 4 of Section II. BC 4: A DAG is use to structure each iniviual agent s knowlege. A DAG moel amits an asymmetric an acyclic interpretation of epenence. Once we aopt it for each agent, we must aopt it for the joint belief of all agents: Proposition 7: Let a multiagent system over constructe following BC through BC 4. Then each subomain is structure as a DAG over an the union of these DAGs is a connecte DAG over. Proof: If the union of subomain DAGs is not a DAG, then it has a irecte loop. This contraicts the acyclic interpretation of epenence in iniviual DAG moels. The connecteness is implie by Proposition 6. The choice 5 of Section II now follows. VII. ON INTERFACE BETWEEN SUBDOMAINS We show that the interface between subomains must be structure as a -sepset (Definition 2). This is establishe below through the concept of -separation [2]. Theorem 8: Let 6 be a hypertree over a irecte graph %$. For each hyperlink? which splits 6 into two subtrees over an be respectively,?>? an?)? are -separate by? if an only if each hyperlink in 6 is a -sepset. Before proving the theorem, we explain its rational an importance. Proposition 6 states that each separator in a tree organization must rener subomains in the two inuce subtrees conitionally inepenent. Because -separation captures all graphically ientifiable conitional inepenencies [2], Theorem 8 implies that -sepset is the necessary an sufficient syntactic conition to ensure conitionally inepenent separators. We prove Theorem 8 below: Proof: [Sufficiency] Assume that each hyperlink is a -sepset. We show that for any given hyperlink?, 7?? an?? are -separate by?. Let be a path between?>? an?>? such that all noes in one sie of belong to 2??, all noes in the other sie belong to?>?, an one or more ajacent noes in? are in between. It suffices to show that every such path is blocke by?. Every has at least one -sepnoe. If one -sepnoe on is tail-to-tail or hea-to-tail, then is blocke by?. Consier the case where has only one -sepnoe. We show that cannot be hea-to-hea on. Suppose

13 3 that is hea-to-hea with parents an on. Because is the only -sepnoe on, neither nor is share by an, say, an. This means 3' 4, an 3' 4,. Because is a -sepnoe, there exists a subgraph 7 that contains. Because 7 is either locate in the subtree over or the subtree over or follows that either 3' 4 hols. Given 3' 4 or 3' 4, either, it must hol: a contraiction. Hence, is either tail-to-tail or hea-totail on. Next, consier the case where contains at least two -sepnoes. We show that one of them cannot be heato-hea on. Pick two -sepnoes an on that are ajacent. Such an o exist accoring to how is efine. The an are connecte either by or by "!. In either case, one of them must be a tail noe. [Necessity] Assume that every hyperlink -separates the two subtrees. We show that each hyperlink is a - sepset by contraiction. Suppose that there exists a share noe such that no subgraph contains (hence not every hyperlink is a - sepset). Then there exists a hyperlink? on 6 an there exist nonempty subsets such that,, 2*, an is incomparable with. Because is incomparable with, there exist but,, an but,. The path "!' between an is renere open by? because is hea-to-hea on. Hence,.?>? an?? are not -separate by? : a contraiction. Theorem 8 implies that -sepset is the necessary an sufficient syntactic conition for conitionally inepenent separators uner all possible subomain structures an observation patterns. We emphasize that -sepset is necessary for the most general case, because by restricting subomain structures (e.g., some agent contains only cause relative to other agents but no effect ) or observation patterns (e.g., some agent has no local observation an only relies on others observations), the -sepset requirement may be relaxe. The choice 6 of Section II now follows. From Propositions 4 an 7, an Theorem 8, the following proposition is implie. Proposition 9: Let a multiagent system be constructe following BC through BC 4. Then it must be structure as a hypertree MSDAG. Proof: From BC through BC 4, it follows that each subomain shoul be structure as a DAG an the entire omain shoul be structure as a connecte DAG (Proposition 7). The DAGs shoul be organize into a hypertree (Proposition 4). The interface between ajacent DAGs on the hypertree shoul be a -sepset (Theorem 8). Hence, the multiagent system shoul be structure as a hypertree MSDAG (Definition 3). VIII. ON BELIEF ASSIGNMENT By Propositions 7, the structure of a multiagent system is a connecte DAG. Hence, a joint probability istribution (jp) over the entire omain can be efine by specifying a local istribution for each noe an applying the chain rule. In a multiagent system, a noe can be internal to an agent or share by two or more agents. The istribution for an internal noe can be specifie by the corresponing agent esigner. When a noe is share, it may have ifferent parents in ifferent agents (e.g., Figure 2 an figure3). Because each share noe is a - sepnoe, Definition 2 implies that for each share variable, there exists a subomain containing all the parents of in the entire omain as state in the following lemma: Lemma 2: Let be a -sepnoe in a hypertree MS- DAG *. Let the parents of in ( be!. Then there exists 7 such that 7" D!. If agents are built by the same esigner, then once 2 7! is specifie for,! for each is implie. If agents are built by ifferent esigners, then it is possible that istributions for a -sepnoe at ifferent subnets may be incompatible with one another. For instance, in Figures 2 an 3, # an # may iffer on CB. We make the following basic commitment for integrating inepenently built agents into a multiagent system: BC 5: Within each agent s subomain, the jp is consistent with the agent s belief. For share noes, the jp supplements each agent s knowlege with others. The key issue is to combine agents belief on a share variable to arrive at a common belief. One iea [2] is to interpret the istribution from each agent as obtaine from a sample ata. The combine 2! can then be obtaine from the combine ata sample. In summary, let agents combine their belief for each share. Then, for each share, let jp be consistent with 2 7!, an for each internal, let jp be consistent with 2! hel by the corresponing agent. It s easy to see that the resultant jp is precisely the one efine in Definition 4, state in the following proposition: Proposition 2: Let a multiagent system be constructe following BC through BC 5. Then the jp over is ientical to that of Definition 4. The last choice of Section II now follows. Pooling Propositions 9 an 2 together, the MSBN representation is entaile by the BCs: in

14 4 Theorem 22: Let a multiagent system be constructe following BC through BC 5. Then it must be represente as an MSBN or some equivalent. Before concluing this section, we emphasize that the belief consistency between agents that is require by BC 5 concerns only the share variables an concerns only the backgroun or prior knowlege of agents about these variables. An agent s belief on private variables are not constraine irectly by the beliefs of any other agents (although it will be influence by what other agents have observe). For the share variables, BC 5 requires only that the agents reach an agreement on the prior belief. At run time, ue to local observations, agents beliefs on share variables can become inconsistent. Bringing their beliefs back to consistency will be achieve by agent communication [29]. IX. CONCLUSION From the following basic commitments: [BC ] exact probabilistic measure of belief, [BC 2] communication by belief over small sets of share variables, [BC 3] a simpler organization of agents, [BC 4] DAG for omain structuring, [BC 5] joint belief amitting agents beliefs on internal variables an combining their beliefs on share variables, we have shown that the resultant representation of a cooperative multiagent system is an MSBN or some equivalent. This result ais comparison with relate frameworks. Multiagent inference frameworks base on efault reasoning (e.g., DATMS [6] an DTMS [8]) o not amit BC, nor oes the blackboar [9]. The BDI architecture [22] has been very influential in builing multiagent systems. It primarily eals with representation of an agent s mental state for practical reasoning [26] although it has been extene to eal with communications between agents [7]. Several frameworks for ecomposition of probabilistic knowlege have been propose. Abstract network [2] replaces fragments of a centralize BN by abstract arcs to improve inference efficiency. Similarity network an Bayesian multinet [6] represent asymmetric inepenence where each subnet shares almost all variables with each other subnet. A neste junction trees [] can exploit inepenence inuce by incoming messages to a cluster an it shares all its variables with the nesting cluster. They were not intene for multiagent systems an o not amit BC 2. Among these alternative frameworks, MSBNs are unique in satisfying both BC an BC 2 in one framework. Junction tree base message passing algorithms, e.g., [9], [24], [4], like the above mentione frameworks for probabilistic reasoning with graphical moels, are not intene for multiagent systems. However, one might interpret a cluster in a junction tree as corresponing to an agent an its subomain. Uner such an interpretation, a junction tree representation satisfys BC an BC 2. However, a cluster correspons to a completely connecte set of variables. There is no internal structure an the agent s belief is essentially represente in terms of a joint probability istribution over its subomain. Clearly, both local inference in an agent an communication among agents will be intractable. Hence a junction tree representation uner a multiagent interpretation oes not amit BC4. On the other han, MSBNs allow an agent s internal knowlege to be encoe as a Bayesian subnet. This allows both local inference within an agent as well as communication to be performe efficiently (when the subnets are sparse). This analysis aresses issues on representational constraints require by MSBNs. In particular, the two key technical constraints, hypertree an -sepset interface, are the consequence of BC an BC 2. Efficient methos for verifying these constraints in a multiagent system have been evelope [32], [3]. One useful consequence of BC 2 an the MSBN framework is that the internal knowlege of each agent is never transmitte an can remain private. This ais construction of multiagent systems by agents from inepenent esigners. Multiagent systems commonly stan in two extremes: self-intereste versus cooperative. MSBNs stan in the mile: agents are cooperative an truthful to each other while the internal know-how is protecte. Reasoning an acting in uncertain omains are essential issues for multiagent systems. A recent tren has focuse on moeling using Markov ecision processes (MDP) [2], [35]. It has been shown [] that the computation for solving istribute MDPs is intractable. Hence, heuristics an approximation must be applie. On the other han, probabilistic inference in sparse MSBNs is istribute, exact, an efficient [29]. Therefore, extening MSBNs to probabilistic reasoning an ecision making over extene time perio (as ynamic Bayesian networks [5] exten BNs) provies an alternative representation to the istribute MDP approach. The istribute MDP approach can be viewe as extening the centralize MDP to multiagent systems. The alternative approach can be viewe as extening multiagent uncertain reasoning from static omains to ynamic omains. The result presente in this paper highlights the role of MSBNs in exploring the alternative approach. Furthermore, our analysis provies guiance to extensions an relaxations of the MSBN framework. Less funamental constraints can be relaxe, e.g., BC 4 so that

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16 6 AUTHORS PHOTO AND BIO Yang Xiang receive the Ph.D. egree from University of British Columbia, Canaa in 992. He was a Postoctoral Fellow at Simon Fraser University, Canaa, Assistant Professor an then Associate Professor in the Department of Computer Science, University of Regina, Canaa, Visiting Professor at Aalborg University, Denmark, an Visiting Professor at University of Massachusetts, Amherst, USA. He is currently an Associate Professor in the Department of Computing an Information Science an Director of Intelligent Decision Support Systems Laboratory, University of Guelph, Canaa. His research interests inclue uncertain reasoning, probabilistic graphical moels, multiagent systems, knowlege iscovery from ata, an business, meicine an inustry applications of intelligent systems. Victor Lesser receive his Ph.D. from Stanfor University in 972. He has been a Professor of Computer Science at the University of Massachusetts at Amherst since 977. He is a founing fellow of AAAI, an the founing presient of the International Founation for Multi-Agent Systems. His major research focus is on the control an organization of complex AI systems. He has been working in the fiel of Multi-Agent Systems for over 25 years. Prior to coming to the University of Massachusetts, he was a research scientist at Carnegie-Mellon University where he was the systems architect for the Hearsay-II speech unerstaning system, which was the first blackboar system evelope.